What each projection optimises
Every projection minimises something
A projection is the solution to an optimisation problem, and naming the objective explains more than naming the family. Some objectives are exact constraints, some are least-squares fits, and one is a table of numbers a man adjusted until it looked right.
Which projection is best
An incomplete question, and the incompleteness is the answer. Every projection preserves something and destroys something else, so the comparison worth making is between a projection and a purpose, not between two projections.
Compromise projections
A projection that preserves nothing exactly can distort everything less than one that preserves something exactly. For a general-purpose world map that is the right trade, and it is why the two most widely used ones today have no exact property at all.
Chebyshev's criterion
The only optimality theorem in the subject. Among all conformal projections of a region, the one with least scale variation is the one whose scale factor is constant on the boundary — a criterion with a proof, a unique answer, and a test anyone can run.
Giving up continuity
Cutting a map into lobes really does reduce the distortion, by a factor that can be measured. What is paid is that the map stops being one surface — and the size of the tear is the number that pictures of interrupted maps never carry.
Fitting the aspect to the region
Choosing a projection is a choice among a few dozen named things. Choosing its aspect is a choice among a continuum, it costs nothing, it changes none of the projection's own properties, and for a long thin country it is worth a factor of 183.
Choosing for a line, not a region
Every criterion in this subject integrates over an area. A pipeline, a railway or a coastal survey is a curve, and the projection an area criterion picks for it is not the one it should have — measurably, by a factor of five thousand.
The average of two projections
The Winkel tripel is literally the arithmetic mean of two other projections, and this site's implementation of it agrees with that mean to the last bit. Averaging beats both ingredients by 26 per cent — and it preserves conformality exactly, destroys equal-area completely, and can turn eighteen per cent of the world inside out without either distortion measure saying so.
How many sheets an atlas needs
A tolerance on the scale error inverts, through Chebyshev's bound, into a sheet radius — and a covering problem turns the radius into a count. One part in a thousand costs 1,210 sheets of 403 kilometres radius, the count goes as the reciprocal of the tolerance exactly, and the projection multiplies it by anything from one to fifty-six.
A projection written as a condition
Instead of a formula, a sentence: the distance from these two places must be exactly right. The map that satisfies it is found by intersecting two circles, it is exact to five parts in a hundred million million, and it exists over the whole sphere for a reason that belongs to the sphere rather than to the construction.
Three conditions are one too many
Two distances fix a point in a plane and a third has no freedom left to be satisfied with. Chamberlin's trimetric construction averages the three positions that satisfy two conditions each, and the spread between them — never zero anywhere, 22 km over North America, growing as the cube of the region — is the price of the extra clause.
A map that cannot be read backwards
Craig's projection answers one question exactly — lay a straight edge from any place to the centre and read the compass course, right to 6 × 10⁻¹⁴ of a degree. It pays by folding: 78°S and 48°S on the same meridian are drawn at the same point, so no inverse exists and nothing else can be read off it at all.
Solving for the map instead of choosing it
Chebyshev's criterion has sat on this site since its second phase with one case it could be applied to: the spherical cap, whose answer is the stereographic projection. For any other region the site stated the criterion and stopped. It is a linear least-squares fit, and the fitted map beats every named projection over the region it was fitted to.
A map with no formula
The solved projection has no name, no formula and no closed-form inverse. It is fourteen numbers — and the rate at which those numbers fall away decides whether a map can be shipped at all: geometrically for a smooth region, and like a power for one with corners.
The aspect has three numbers, not one
The site's aspect search has swung the projection's axis through one plane for a long time, and recorded that a region whose long axis runs diagonally has its optimum somewhere that plane never reaches. Searching the whole sphere of pole positions finds 2.6 times more improvement over Japan and 3.4 over the conterminous United States.
The sphere is not the plane at small counts
The site's atlas arithmetic multiplies an ideal sheet count by 2π/√27, the thinnest covering density of the plane. At the four counts whose optimal covering of the sphere is a theorem the plane's number is 21 per cent high at two caps and 9, 5 and 2 per cent low at four, six and twelve — wrong in both directions, and the direction changes with the count.
The third parameter, run
An aspect has three numbers and this site has been searching two of them, with a note admitting it. Searching all three is worth up to 2.1 times — and the obvious way to do it, starting from the two-parameter answer and letting the third move, finds a fraction of that or nothing at all.
Every equal-area map is every other one
Take Mollweide and slide every row of the page sideways by an amount that depends on the row. The result satisfies the equal-area condition to 6 × 10⁻¹², exactly as well as Mollweide does, and it is a ruin — the angular deformation at one ordinary point has gone from 11° to 60°. Equal-area is one equation, and one equation leaves a whole function free.
The landscape the search walks on
The three-parameter aspect search was run and its answer recorded with a note admitting nothing proved it global. Mapping the objective finds 26 to 34 local minima for every projection and region tried, a downhill walk from a random start reaching the best of them 6 to 35 per cent of the time — and one seed from the coarse grid the search already uses reaching it in all four cases. The score is reproducible to two per cent across a sevenfold refinement; the pole it names moves 60 degrees.
Report the map, not the parameters
The aspect search was found returning the same score to 2.3 per cent from poles sixty degrees of latitude apart, which left the answer looking irreproducible. That reading assumed the disagreeing triples make disagreeing maps. They do not — the three answers agree on the distortion field to a quarter of the deformation the map already has.
Not every distortion can be asked for
Six essays have written projections as conditions and asked how much freedom a condition leaves. The reverse question has never been put: a cartographer knows what distortion they want, so can they ask for it? For a conformal map the answer is a single equation, it is the Theorema Egregium in disguise, and asking for no distortion anywhere fails it by exactly the curvature of the sphere.
The shape of the valley
An aspect search returns three numbers, two searches return triples that differ by a hemisphere, and the maps they produce agree. One cause is an exact degeneracy and the rest was called a valley and left unmeasured. Sampled densely, it is neither a valley nor a basin: a connected sheet spanning 170° of pole that fractures into fourteen pieces once the threshold tightens.
Where the valley breaks in two
A near-optimal set is one connected sheet at a loose threshold and fourteen basins at a tight one, and the transition was explained without being tested. The explanation is a prediction about region size and projection sharpness: swept over both, the threshold falls from 2.13 to 0.27 as a region grows from 6° to 30°, three projections lie on nearly one curve, and a fourth declines to join for a reason worth having.
The nearest map to an impossible request
Rung seven found that not every distortion can be asked for. It never asked what happens when one is asked for anyway — and the answer has a shape: the achievable fields are the solutions of an elliptic equation, a request is a point off that set, and the nearest point leaves a residual whose floor is the curvature rather than the size of the ask.
The height of the pass between two basins
One thing was left owed: the fracture threshold falls as the −1.33 power where the argument predicts −1, and the difference was supposed to be the height of the pass. It is not. Measuring the pass directly gives −1.60, which is further from the prediction, and the two basins turn out to be exact symmetric copies with nothing to decompose.
The nearest equal-area map to an impossible request
Every number in the measurement before this one is inside the conformal achievable set, because Liouville's is the conformal condition. The equal-area set is one equation on two functions and never refuses: the same four requests are met to nine parts in a billion, and charged for in angle instead.
The basins have widths as well as depths
Measuring the height of the pass left one thing owed: shape means widths too. Measured, the basin has three of them — 32°, 16° and 7° at Japan — it gets wider rather than narrower as the region grows, and the exponent it predicts overshoots the measured one by half again.
The threshold is not a percolation
The near-optimal aspect set was found breaking into twelve pieces rather than two, the transition was called a percolation, and the exponent went unmeasured. Swept finely, the piece count rises from one to twenty-three and falls back to one — and refining the grid by a factor of fifteen does not move the peak, while an uncorrelated field on the same lattice grows by a factor of twelve.
A condition imposed at points is not a condition
Nine rungs state a condition and solve it, and every solve imposes the condition at a finite set of samples because that is what a linear system is. With barely more equations than unknowns the residual the solver reports is 8.3 times too good — and refining the collocation twentyfold does not improve the map at all, it only makes the report honest.
The first break is mostly its denominator
Three earlier measurements have fitted the near-optimal set's fracture threshold against region size and read the answer as a statement about the landscape. It is a ratio, and separating it takes one multiplication: the pass's own depth is constant to 12 per cent below twenty degrees of span, and the whole of the threshold's movement there is the denominator — the best score the region admits at all — rising with exponent 0.92.
The nodes were evenly spaced
Refining an evenly collocated fit improves the solver's report and leaves the map alone. Moving the same number of nodes to the Chebyshev positions — crowded toward the corners, where a conformal map of a polygon is singular — makes the fit settle: 7.030 × 10⁻⁵ at forty-eight nodes and exactly that at every count above it, against an even fit that wanders by 43 per cent and never converges at all.
The pooled score abandons a region
Fifteen rungs optimise for one region. An atlas is several, and pooling their samples into one area-weighted score is what everybody does — which on Britain and New Zealand serves Britain 1.2 times worse than it could be served alone and New Zealand 125 times worse. The worst-case objective makes them equal at 33 and 59, and the cost of sharing rises with separation from 1.4 to 59.
Chebyshev's map is the best at its worst, and not on average
Chebyshev's criterion makes a conformal map's scale constant on the boundary of a region, and no conformal map has a smaller range of scale inside it. The same series, fitted instead to make the scale as nearly constant as possible over the area, is a different map on every region but a cap. On a spherical square Chebyshev's map is 8.3 per cent worse in root-mean-square, the other map's range is 16.5 per cent wider, and every map between them trades one for the other.
The map that keeps the most ground inside a tolerance
A surveyor does not want the smallest worst case or the smallest mean square. They want as much of the territory as possible within a stated tolerance of true scale and are indifferent to how far outside the rest goes. That is a third criterion, it is answered by a third map, and on a 25° square at half a per cent it keeps 60.1 per cent of the area where the mean-square map keeps 45.4 and Chebyshev's 36.5 — bought by sending a tail out to 5.7 per cent where Chebyshev's worst is 3.0.
A criterion worth using is one whose answer is not unique
Forty-two starts on one region and one tolerance give forty-two different maps, keeping from 36.70 per cent of the area to 65.15. The two starts anybody would actually use — Chebyshev's map and the mean-square map — end 22.7 points apart, which is more than the criterion buys over either of them. And the spread is widest at exactly the tolerance where the criterion is worth using at all, because both facts are the same fact: an objective with a flat top and a sharp edge has somewhere to hide answers.
The pass that fails first is not the one that was measured
Three attempts to explain why the near-optimal aspect set comes apart when it does have fitted the height of the pass between the two deepest basins. That is not the pass that fails first: at every region size the set is held together by a shallow pair far from the optimum, standing 1.5 to 9.8 times higher, and the two fall at quite different rates — the ‑0.60 power of the region's span against the ‑1.31. And the piece count the older instrument bisects on is not monotone, so there was no single crossing to find.
The constant belongs to the projection, not to the problem
Separating the fracture threshold into a pass depth and the best score a region admits left the pass depth constant below twenty degrees of span and nobody saying what constant. Measured across three projections and four latitudes it is not one number: it runs over a factor of 5.7 while the denominator blamed for the movement runs over 1.5. So the reading reverses with what is varied — over region sizes the numerator is fixed and the denominator moves, and over projections and latitudes it is the other way round.
37 essays in this field, the first 16 of them shown with their opening figure.